| Summary: | abstract: A signal with time-varying frequency content can often be expressed more clearly using a time-frequency representation (TFR), which maps the signal into a two-dimensional function of time and frequency, similar to musical notation. The thesis reviews one of the most commonly used TFRs, the Wigner distribution (WD), and discusses its application in Fourier optics: it is shown that the WD is analogous to the spectral dispersion that results from a diffraction grating, and time and frequency are similarly analogous to a one dimensional spatial coordinate and wavenumber. The grating is compared with a simple polychromator, which is a bank of optical filters. Another well-known TFR is the short time Fourier transform (STFT). Its discrete version can be shown to be equivalent to a filter bank, an array of bandpass filters that enable localized processing of the analysis signals in different sub-bands. This work proposes a signal-adaptive method of generating TFRs. In order to minimize distortion in analyzing a signal, the method modifies the filter bank to consist of non-overlapping rectangular bandpass filters generated using the Butterworth filter design process. The information contained in the resulting TFR can be used to reconstruct the signal, and perfect reconstruction techniques involving quadrature mirror filter banks are compared with a simple Fourier synthesis sum. The optimal filter parameters of the rectangular filters are selected adaptively by minimizing the mean-squared error (MSE) from a pseudo-reconstructed version of the analysis signal. The reconstruction MSE is proposed as an error metric for characterizing TFRs; a practical measure of the error requires normalization and cross correlation with the analysis signal. Simulations were performed to demonstrate the the effectiveness of the new adaptive TFR and its relation to swept-tuned spectrum analyzers. === Dissertation/Thesis === M.S. Electrical Engineering 2012
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